chapter2(3) - 121 MATH1010 Chapter 2 cont… 3 Application...

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MATH1010: Chapter 2 cont… 1 LIMITS AND DERIVATIVES cont… Precise Definition of a Limit (Section 2.4, pg. 114) cont… Recall: Last day, we considered the function = + = 1 1 , 5 , 7 2 ) ( x x x x f and wondered what it really meant to say that 9 ) ( lim 1 = x f x here. Definition: Let f be a function defined on some open interval that contains the number a , except possibly at a itself. Then we say that the limit of ) ( x f as x approaches a is L , and we write L x f a x = ) ( lim if for every number 0 > ε there is a number 0 > δ such that if < < a x 0 then < L x f ) (
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MATH1010: Chapter 2 cont… 2 Example: Prove that 14 1 5 lim 3 = x x Is the choice of δ unique? One can also do precise definitions for sided limits in a similar way…refer to definitions 3 and 4 on page 118 of your text (the only difference is the inequality for x ). Also, infinite limits can be defined formally in a similar manner…refer to definition 6 on page
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Unformatted text preview: 121. MATH1010: Chapter 2 cont… 3 Application: According to Ohm’s law, when a voltage of V volts is applied across a resistor with a resistance of R ohms, a current of R V I = amperes flows through the resistor. Suppose that the voltage remains constant at 3.0 volts, but temperature variations cause the resistance to vary from a value of 7.5 ohms. a) If the current is not allowed to vary by more than 001 . ± ampere, what variation of δ ± from the value of 7.5 ohms is allowable for the resistance? b) In the context of the delta-epsilon definition of L x f a x = → ) ( lim , what are x , ) ( x f , a , L , ε , and ? [Source: Modified from “Calculus: Early Transcendentals” 8 th ed. by H. Anton, I. Bivens, and S. Davis, 2005] Example:...
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chapter2(3) - 121 MATH1010 Chapter 2 cont… 3 Application...

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